WASHINGTON (AP) — The Senate is preparing to push landmark immigration legislation across a few final hurdles and possibly pick up one or two more Republican “yes” votes along the way.

The legislation cleared an early procedural hurdle Monday with room to spare, and more test votes awaited Wednesday. The White House-backed bill would pour billions of dollars into border security and offer a path to citizenship to some 11 million immigrants now in the United States illegally.

After the addition of provisions greatly strengthening border security and doubling the size of the border patrol, the legislation looks likely to command support from at least 14 Republicans on final passage Thursday or Friday. That’s more than enough to ensure the 60-vote margin needed for passage, as all 52 Democrats and the two independents who usually vote with them look likely to stick together.

Some supporters would like to bring even more Republicans on board and are working to negotiate votes on amendments to win over anyone who is wavering.

Sen. Rob. Portman, R-Ohio, is pushing an amendment to strengthen an electronic employment verification program made mandatory in the bill.

“I can’t vote for (the bill) without it,” Portman said.

Sen. Saxby Chambliss, R-Ga., wants changes to a new agriculture workers program that he says makes it too easy for farm workers to get permanent U.S. residency.

The measures sought by Portman and Chambliss are being opposed by some immigrant advocacy groups, and some Senate Democrats believe the bill has enough Republican support as is without pursuing more changes. But Sen. Chuck Schumer, D-N.Y., a lead sponsor of the bill, continues to push for further amendments, arguing that a large bipartisan vote in the Senate is necessary to pressure the GOP-controlled House to act.

In the House on Wednesday, the Judiciary Committee was to vote on its own employment verification legislation. It’s the third in a series of single-issue immigration bills the committee has voted on as it takes a piecemeal approach to overhauling the nation’s immigration system, in contrast with the Senate’s comprehensive bill.

None of the measures considered by the House panel includes a path to citizenship or even legalization for the millions here illegally, something opposed by many House conservatives. That leaves the final fate of the immigration overhaul effort unclear once the spotlight moves from the Senate to the House after this week’s action.